EGA1: The Language of Schemes | The n-Category Café
WEBFeb 6, 2020 · EGA1: The Language of Schemes. Posted by John Baez. Tim Hosgood, Ryan Keleti and others have finished an English translation of this classic: Alexander Grothendieck, EGA1: The Language of Schemes. and the LaTeX files are now open-source on GitHub. They are working on the rest of EGA, and they could use help! Posted at February 6, 2020 1:28 AM UTC.
No New Normed Division Algebra Found! | The n-Category Café
WEBOctober 20, 2020. No New Normed Division Algebra Found! Good news! The paper mentioned in my last article here, Eight-dimensional octonion-like but associative normed division algebra, has been retracted: Scott Chapman, Statement of retraction: Eight-dimensional octonion-like but associative normed division algebra, Communications in …
Enrichment and the Legendre-Fenchel Transform II
WEBMay 22, 2014 · The ˉℝ -distance is a refinement of this partial order in the f1 ≥ f2 if and only if d(f1, f2) ≤ 0. The standard fact that the Legendre-Fenchel transform is order reversing is then a corollary of the above corollary: in other words we get that for convex functions f1, f2: V → ˉℝ we have. f1 ≥ f2 ⇔ f * 2 ≥ f * 1.
Magnitude Homology Equivalence | The n-Category Café
WEBJun 18, 2024 · We take the third option, defining metric spaces X and Y to be magnitude homology equivalent if there exist maps X ⇆ Y whose induced maps Hn(X) ⇆ Hn(Y) are mutually inverse for all n ≥ 1. Here “map” means a map that is 1-Lipschitz, or short, or a contraction or distance-decreasing in the non-strict sense. When you view metric spaces ...
Categorical Homotopy Theory | The n-Category Café
WEBJun 7, 2014 · In my first year at Harvard, I had an opportunity to teach a graduate-level topics course entitled “Categorical Homotopy Theory.”. Its aim was to highlight areas in which category theoretic abstractions provide a particularly valuable insight into classical homotopy theoretic constructions. Over the course of the semester I gave lectures ...
Optimal Transport and Enriched Categories II
WEBJun 19, 2021 · The dual transport problem from general linear programming duality. Last time I described the basic, discrete optimal transport problem in which there are some suppliers wanting to supply certain amounts of some resource (eg loaves of bread) and some receivers wanting to receive a certain amount of that resource. There’s a cost associated to transporting a unit of the resource from a ...
Inevitability in Mathematics | The n-Category Café
WEBJun 28, 2010 · Inevitability in Mathematics. I’m just back from a conference in Nancy – From Practice to Results in Logic and Mathematics. As you see from the abstract, I was speaking about the inevitability of certain things appearing in mathematics: entities, facts, theories, ideas. We had a discussion of the related robustness a while ago.
The Modular Flow on the Space of Lattices
WEBApr 7, 2014 · Now set L ≔ P(ℤ2). We claim that L is a periodic orbit of period t. Indeed: Lt = (et 0 0 e − t)P(ℤ2) = ± PA(ℤ2) = ± P(ℤ2) = L. We have just proved one direction of the following. Theorem: The periodic orbits of the modular flow are in bijection with the conjugacy classes of hyperbolic elements in SL (2, ℤ).
Line Bundles on Complex Tori (Part 2) | The n-Category Café
WEBLast time I explained how the job of classifying holomorphic line bundles on a complex torus X X breaks into two parts:. the ‘discrete part’: their underlying topological line bundles are classified by elements of a free abelian group called the Néron–Severi group NS (X) \mathrm{NS}(X).. the ‘continuous part’: the holomorphic line bundles with a given underlying topological line ...
Screw Theory | The n-Category Café - University of Texas at Austin
WEBOct 21, 2019 · Screw theory is about the Euclidean group: the group of rigid motions of Euclidean space. A screw is an element of the Lie algebra of this group. It’s a 6d vector built from a pair of 3d vectors: an infinitesimal translation and an infinitesimal rotation. An object moving through space and rotating has a velocity and an angular velocity.